MinT - Best Known (34−13, 34, s)-Nets in Base 8

Best Known (34−13, 34, s)-Nets in Base 8

(34−13, 34, 208)-Net over F₈ — Constructive and digital

Digital (21, 34, 208)-net over F₈, using

2 times m-reduction [i] based on digital (21, 36, 208)-net over F₈, using
- trace code for nets [i] based on digital (3, 18, 104)-net over F₆₄, using
  - net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
      - a function field by SÃ©mirat [i]

(34−13, 34, 300)-Net in Base 8 — Constructive

(21, 34, 300)-net in base 8, using

trace code for nets [i] based on (4, 17, 150)-net in base 64, using
- 4 times m-reduction [i] based on (4, 21, 150)-net in base 64, using
  - base change [i] based on digital (1, 18, 150)-net over F₁₂₈, using
    - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 1 and N(F) ≥ 150, using
        a function field by SÃ©mirat [i]

(34−13, 34, 353)-Net over F₈ — Digital

Digital (21, 34, 353)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8³⁴, 353, F₈, 13) (dual of [353, 319, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(8³⁴, 511, F₈, 13) (dual of [511, 477, 14]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 511Â = 8³âˆ’1, defining interval IÂ =Â [0,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]

(34−13, 34, 39635)-Net in Base 8 — Upper bound on s

There is no (21, 34, 39636)-net in base 8, because

1 times m-reduction [i] would yield (21, 33, 39636)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 633892 119934 677630 915501 824716 > 8³³ [i]